Mean-field limit for a class of stochastic ergodic control problems

Abstract

We study a family of McKean-Vlasov type ergodic optimal control problems with linear control, and quadratic dependence on control of the cost function. For this class of problems we establish existence and uniqueness of optimal control. We propose an N -particles Markovian optimal control problem approximating the McKean-Vlasov one and we prove the convergence in total variation of the law of the former to the law of the latter when N goes to infinity. Some McKean-Vlasov optimal control problems with singular cost function and the relation of these problems with the mathematical theory of Bose-Einstein condensation is also discussed.

A stochastic control approach to Sine Gordon EQFT

“…Below we will show that α(t) t β 2 /8π and J t u L p t −1 u L p so one can hope for the functioal to be convex provided that λ is small enough and u stays in a bounded region, which is guaranteed by Theorem 1. Expaning this heuristic we will prove (2), which in turn will allow us to remove the cutoff ρ in the variational formula and prove the following theorem:…”